UNIVERSITA’ DI BOLOGNA
SCHOOL OF ENGINEERING
-Forlì
Campus-SECOND CYCLE MASTER’s DEGREE in
AEROSPACE ENGINEERING
Class: LM-20
GRADUATION THESIS in:
Spacecraft Attitude Dynamics and Control
Design of Thrust Vectoring attitude
control system for Lunar Lander
flying testbed
CANDIDATE
SUPERVISOR
David Bernacchia
Prof. Marco Zannoni
CO-SUPERVISOR
Prof. David Barnhart
V Session
The proposed work has been developed within the project LEAPFROG (Lunar En-try and Approach Platform For Research On Ground) at the University of Southern California. The project concerns the realization of a lunar lander test bed prototype with the aim of testing GNC algorithms for simulated lunar flight and descent. The main focus is the realization of a newly designed thrust vectoring system (TVC) that exploits the thrust given by a main engine in order to control the attitude of the platform. This new attitude control system is combined with current tradtional reaction control system (RCS) based on cold-gas thrusters. After a preliminary hardware design phase, a linear LQR controller, based on a reduced quaternion model, and a non-linear sliding mode controller are designed for the TVC system. Linear Quadratic Regulator offers a simple implementation, an optimal control law. However it can be affected by un-modeled dynamics and the solutions provided are, in general, only locally valid. Sliding mode control (SMC) guarantees robustness against disturbances, unmodeled dynamics and uncertainties about the mass prop-erties of the prototype, offering also a global stability. Cons of this method are the hard implementation and the request of an high-frequency actuation. A MAT-LAB/simulink simulation is set up in order to validate and compare the designed controllers and to analyze if the thrust vectoring system leads to the desired results.
1 Introduction 1
1.1 Introduction to LEAPFROG project . . . 1
1.2 Generation 2: proposal and goals . . . 3
1.3 Thesis work and outline . . . 4
2 Theoretical background 7 2.1 Landing on a celestial body . . . 7
2.2 Frames of reference . . . 9
2.3 Attitude representation . . . 10
2.3.1 Euler Angles representation . . . 11
2.3.2 Quaternions representation . . . 12
2.4 Attitude kinematics . . . 13
2.5 Rigid body attitude dynamics . . . 15
2.6 Thrust vector control: literature review . . . 17
2.6.1 Reactive fluid injection . . . 17
2.6.2 Exhaust flow deflection . . . 18
2.6.3 Engine mechanical manipulation . . . 19
3 Preliminary hardware design 23 3.1 Project budget and requirements . . . 23
3.2 Structure . . . 23
3.2.1 Chassis . . . 25
3.2.2 Legs . . . 26
3.2.3 Platforms . . . 26
3.3 Propulsion system . . . 27
3.4 Reaction control system . . . 28
3.5 Thrust Vector Control system . . . 30 v
vi CONTENTS
3.5.1 Gimbal joint design . . . 33
3.5.2 Actuation system . . . 35
3.5.3 Embedded electronics . . . 39
4 Control system design 41 4.1 System Dynamics . . . 41
4.2 Linear Quadratic Regulator . . . 44
4.2.1 Cost function and Riccati equation . . . 45
4.2.2 Reduced quaternion model . . . 47
4.2.3 Asymptotic stability . . . 49
4.3 Sliding Mode Control . . . 50
4.3.1 Sliding manifold and control law definition . . . 51
4.3.2 Chattering phenomenon . . . 53
5 Control system simulation 57 5.1 Input parameters . . . 57
5.2 LQR controller simulations . . . 58
5.2.1 Ideal case . . . 60
5.2.2 Perturbed case . . . 62
5.3 Sliding mode controller simulations . . . 72
5.3.1 Ideal case . . . 73
5.3.2 Perturbed case . . . 75
Conclusions and future work 84
Appendix A 85
Bibliography 88
Figure 1.1 The Lunar Landing Research Vehicle . . . 2
Figure 1.2 LEAPFROG generation 1 (CAD model) . . . 3
Figure 2.1 Apollo lunar landing sequence [7] . . . 8
Figure 2.2 Frames of reference adopted . . . 9
Figure 2.3 Reactive flow injection on TitanIIIE-Centaur (NASA) [1] . . . 18
Figure 2.4 Example of gimbaled thrust on a rocket [6] . . . 19
Figure 2.5 Typical joints used on gimbaled engines [10] . . . 20
Figure 2.6 LLRV gimbaled engine [20] . . . 21
Figure 3.1 Structure of Generation 2 - CAD model (front view) . . . 24
Figure 3.2 Structure of Generation 2 - CAD model (top view) . . . 24
Figure 3.3 Carbon fiber chassis - final version . . . 25
Figure 3.4 JetCat P300 PRO [2] . . . 27
Figure 3.5 RCS thrusters placement . . . 28
Figure 3.6 RCS block diagram . . . 30
Figure 3.7 Conceptual sketch of the gimbal ring . . . 33
Figure 3.8 Gimbal ring mechanical design . . . 34
Figure 3.9 Windynation linear actuator [33] . . . 35
Figure 3.10 Actuators: speed vs load [33] . . . 36
Figure 3.11 Actuators installation scheme . . . 38
Figure 3.12 Thrust vectoring system embedded electronics . . . 39
Figure 3.13 LEAPFROG Generation 2 . . . 40
Figure 4.1 Thrust vector decomposition . . . 43
Figure 4.2 Ideal sliding mode [13] . . . 50
Figure 4.3 Chattered sliding mode [12] . . . 54
Figure 5.1 Pitch angle control . . . 60
Figure 5.2 Pitch rate control . . . 61
Figure 5.3 Gimbal offset angle δ . . . 61 vii
viii LIST OF FIGURES
Figure 5.4 Gimbal rotation angle ξ . . . 62
Figure 5.5 Half-sine unitary impulse . . . 63
Figure 5.6 Pitch channel . . . 63
Figure 5.7 angleδ - impulse response (pitch) . . . 64
Figure 5.8 angleδ - impulse response (pitch and yaw) . . . 64
Figure 5.9 angleξ - impulse response (pitch and yaw) . . . 65
Figure 5.10 Yaw channel - external random noise . . . 66
Figure 5.11 Pitch channel - external random noise . . . 66
Figure 5.12 Angle δ in presence of random noise . . . 67
Figure 5.13 Angle ξ in presence of random noise . . . 67
Figure 5.14 Displacement linear actuator 1 . . . 68
Figure 5.15 Displacement linear actuator 2 . . . 68
Figure 5.16 Yaw channel - presence of state measurements errors . . . 69
Figure 5.17 Pitch channel - presence of state measurements errors . . . 69
Figure 5.18 Pitch angle - different scenarios . . . 70
Figure 5.19 Yaw angle - different scenarios . . . 71
Figure 5.20 Gimbal offset angle - different scenarios . . . 71
Figure 5.21 Chattered phase plane . . . 73
Figure 5.22 Pitch angle and pitch rate behaviour under sliding mode control 74 Figure 5.23 Angle δ under sliding mode control . . . 74
Figure 5.24 Pitch channel - impulse disturbance . . . 75
Figure 5.25 Yaw channel - impulse disturbance . . . 75
Figure 5.26 Gimbal offset angle - combined pulses disturbances . . . 76
Figure 5.27 Gimbal rotation angle - combined pulses disturbances . . . 76
Figure 5.28 Displacement linear actuator 1 - pulse disturbance . . . 77
Figure 5.29 Displacement linear actuator 2 - pulse disturbance . . . 77
Figure 5.30 Pitch and yaw angles - random disturbance . . . 78
Figure 5.31 Displacement linear actuator 1 - random disturbance . . . 78
Figure 5.32 Displacement linear actuator 2 - random disturbance . . . 79
Figure 5.33 Pitch angle fluctuation due to measurements errors . . . 79
Figure 5.34 Pitch angle - different scenarios . . . 80
Figure 5.35 Yaw angle - different scenarios . . . 80
Introduction
1.1
Introduction to LEAPFROG project
In the last few years, lunar exploration has been experiencing a new "golden age", after the intense period of interest from the well-known Apollo missions. News from the recent Chandrayaan-2 mission by Indian Space Research Organisation and NASA’s 2024 goal to bring back humans to the Moon’s surface, Moon exploration and lunar landing are of great interest not only for big national or international agencies, like NASA or ESA, but also for private companies, research centers and universities. University of Southern California’s (USC) Space Engineering Research Center (SERC), with the support from the Information Sciences Institute (ISI), started its personal research in prototype lunar lander technology in 2006, with the birth of the LEAPFROG project.
LEAPFROG (Lunar Entry and Approach Platform For Research On Ground) is a lunar lander prototype built by students with the aim of simulating lunar gravity on Earth. Since the very beginning, the main goal of the project has been the development of a test bed lunar prototype vehicle that can fly multiple times in Earth’s gravity through free flight to simulate a lunar descent and landing sequence. The concept was inspired by NASA’s Lunar Landing Research Vehicle (LLRV), created to investigate and analyze different piloting techniques that could be used for the descent and landing of the Apollo Lunar Module [5]. The LLRV project, in fact, was started even before NASA had selected the landing strategies to use for the Lunar Module (LM). The LLRV led to the Lunar Landing Training Vehicle (LLTV), a vehicle designed with characteristics closer to the planned LM, to better represent the final descent phase, used for a more advanced training of the Apollo astronauts [20].
2 INTRODUCTION TO LEAPFROG PROJECT
Figure 1.1: The Lunar Landing Research Vehicle
From the first day, the LEAPFROG project was planned to be a multi-generational activity, where every new academic year was intended to increase the performance and capability of the simulated lander through each subsequent new generation of the vehicle. The first prototype (Generation 0) used all commercial off the shelf hardware, in order to maintain the lowest possible cost and to allow students to have a true “hands on” training in the system life cycle of a simulated space vehicle [5]. It was characterized by a a kerosene-powered JetCat P200 jet engine used for hover and descent flight, while lateral and rotational control was provided by a Reaction Control System (RCS) composed by 12 cold-gas thrusters driven by solenoid valves which regulated pulse of air with an on-off configuration. Guidance, Navigation and Control (GNC) codes were then developed in order to maintain the platform self-equilibrium, using stability measurements coming from the primary avionics sensor, an inertial measurement unit (IMU) [16]. At the end of the design and building phase, some flight tests were performed to address the behaviour of the prototype. Even if the flight tests showed some encouraging results, many issues were encountered, primarily related to a non-complete optimization of the vehicle. A new phase for LEAPFROG started back up in 2018, starting from the goals and end results obtained in Generation 0, to build a new generation of the vehicle (Generation 1) with improved capabilities, to fix the main issues encountered with the first prototype. One of the main problems of the first vehicle was the difficulty to maintain a stabilized configuration in hover flight. For this reason the main focus of Generation 1 team was the improvement of the cold-gas thrusters based attitude control system, testing different sizes for the thrusters nozzle and designing new arms capable to increase the control torque given by the thrusters themselves. The general structure was kept almost equal to the one of the previous generation
since particular requirements in terms of payload capacity were not given for this prototype.
Figure 1.2: LEAPFROG generation 1 (CAD model)
1.2
Generation 2: proposal and goals
In September 2019, a new team further improved the design and subsystems of the LEAPFROG, starting Generation 2 of the project. The work, focused on additional improvements to the new generation prototype,with additional requirements:
• Increase payload capacity
• Increase thrust to weight ratio
• Increase the flight time
• Enable a more re-configurable structure
• Enable capability to withstand a free fall from up to 3 meters
• Optimize and improve subsystems and guidance algorithms
• Continue to pursue lowest possible cost
4 THESIS WORK AND OUTLINE
Secondary goals were also considered for a future vehicle design that would be closer to an actual lunar lander prototype, by transitioning the current jet engine with a rocket propulsion system and integrating space-qualified hardware [5].
Starting from the primary requirements, a new structure and subsystems were de-signed and developed, with a planned flight test scheduled for the end of Spring 2020.
1.3
Thesis work and outline
The present thesis work was developed within the Generation 2 of the LEAPFROG project. Starting from the purpose of improving the subsystems relative to the attitude control and increase the payload capacity of the prototype, my research work concerned the design and implementation of a merged multi-control GNC system based on a newly designed thrust vector control (TVC) system, that ties into the air breathing turbine engine, that could work together with the old RCS based on cold-gas thrusters. The main idea behind the TVC system is to exploit the energy given by the main engine in order to control and balance the platform. The principal hardware of the system is characterized by a gimbal ring that allows the motion of the principal engine around a particular center of rotation. The engine is moved and oriented to the desired position by a couple of linear electro-mechanical actuators. Recalling the view of a change of the propulsion system, the design of the TVC system should work with different thrust outputs (i.e jet engine or rocket engine) and should be adaptable to different thrust values and environmental conditions. The aim of the work was to verify if, through the implementation of the TVC system, it’s possible to avoid, or at least reduce, the usage of the cold-gas thrusters system, thus reducing the consumption of RCS gas. In fact, the implementation of the thrust vectoring system, not only can improve the maneuverability of the prototype, but it can also lead to a reduction of the total mass of the structure. This happens because, with an important reduction of RCS propellant consumption, we may be able to reduce the carried weight relative to the cold-gas tanks, saving mass and, consequently, increasing the payload capacity of the vehicle.
In parallel with the work on the TVC system, I collaborated, with the other team members, in the development and design of the new structure and other subsystems of the prototype.
From here on, the thesis will be composed by 4 more chapters divided as follows: the 2nd chapter will give an introduction to all the theoretical concepts behind the
analysis of a rigid body’s attitude and the development of an attitude control system. The 3rd chapter will describe the design phase of the new prototype, including
both the general structure and, more in detail, the hardware relative to the Thrust Vector Control system. In the 4th chapter the dynamic model of the prototype will be write down in terms of equations of motion and the theoretical design of the control system algorithms will be presented. The 5th and last chapter will show the
results obtained implementing the control systems described in the 4th chapter, in a
MATLAB/Simulink environment. Final conclusions will be discussed, together with the expected future works that can be done to improve the Guidance Navigation and Control system of the LEAPFROG, relative to the main goals of the project.
Theoretical background
In the present chapter are presented the theoretical foundations on which the prob-lem of attitude representation, determination and control is based. A quick review and introduction to the concepts of Thrust Vector Control and landing sequence are also given.
2.1
Landing on a celestial body
When we talk about lander spacecraft, we are referring to particular spacecrafts that are designed to reach the surface of a celestial body and be able to land successfully in order to communicate with Earth [22]. The design of the landing sequence is a crucial task for the success of a mission and the strategies used are strictly related to the characteristics and the nature of the target body and the lander. For example, a landing performed on a planet with an atmosphere would be characterized by different strategies with respect to a landing maneuver performed on an airless body, like an asteroid or the Moon. In the first case, in fact, the interaction between the atmosphere and the probe can be exploited to slow down the latter through the usage of different devices like parachutes. In the case of an airless body, since this method cannot clearly be used, the slowdown is entrusted to the propulsion system and is usually combined to other particular maneuvers. A very simple descent control strategy is the so called gravity turn in which the thrust vector is aligned with the velocity vector in order to cancel the initial horizontal component of the spacecraft’s velocity. The continuous braking, combined with the action of the gravity, progressively changes the velocity vector, which becomes more and more dominated by the vertical component induced by the gravity itself. The simplicity of this particular maneuver is counterbalanced by its poor behaviour in terms of fuel consumption. This because, being a quite slow maneuver, the more it takes the landing the more a vertical velocity component, that has to be cancelled out by the
8 LANDING ON A CELESTIAL BODY
propulsion system, is added [4]. This particular landing strategy is called powered descent. A typical example of soft landing sequence in Moon environment is the one adopted by the Apollo missions, depicted in Figure 2.1:
Figure 2.1: Apollo lunar landing sequence [7]
As we can see, the full maneuver is characterized by three different phases: braking phase, approach phase and landing phase. The braking phase is the first part of the powered descent and it’s designed to reduce the orbital velocity of the spacecraft and, consequently, the altitude. The approach phase was designed for the pilot in order to monitor the approach to the surface of the Moon. The final landing phase starts when the altitude is strongly reduced (around 150 meters in the case of Apollo mission [7]) and should be performed with a low velocity and an attitude such that a soft vertical landing is guaranteed. During the whole landing maneuver, the attitude control is really important since the transition between the different phases should be, in terms of attitude angles, as smooth as possible.
However, thinking about lunar exploration, is clear that the possibility to analyze and have surface measurements in different locations is scientifically interesting. For this reason landers are usually designed to be able to hover and perform a so called powered re-ascent, in order to move to different spots. Without atmosphere, this may be the only way to travel long distances over the lunar surface efficiently and quickly. This type of landers that are able to take off vertically, travel and land softly again on lunar surface, are defined as Vertical Take-off Vertical Landing spacecrafts (VTVL) [7]. The LEAPFROG prototype conceptually belongs to this category of vehicles since one of its goal will be being able to perform multiple hops, landing softly each time.
In general the vertical soft landing task is much harder to achieve with respect to the vertical take-off or the travelling case, due to the many constraints in terms of
attitude angles and attitude angular velocity. This underlines the importance of the attitude control system while the landing sequence is performed.
2.2
Frames of reference
Since the work concerns the development of an attitude control system for a pro-totype performing a landing maneuver, the frames of reference considered are the following:
– Body-centered frame of reference
The origin coincides with the center of mass of the body while the axis are directed along the principal axis of inertia. It is described by the triad {ˆi,ˆj,ˆk}
– Planet-fixed frame of reference
This frame of reference is fixed to the planet on which the landing is performed (the Earth for LEAPFROG). The origin is placed in correspondence of the target landing spot, the c3ˆ axis is perpendicular to the surface and aligned with the center of the Earth, the other 2 axis lay on the surface, pointing
ˆ
c1 towards east and cˆ2 towards north, relative to the magnetic north. The
short amount of time that takes the landing maneuver to be done allows us to neglect the rotation of the Earth and consider this frame of reference inertial. It is described by the triad {cˆ1,cˆ2,cˆ3}
The frames of reference are depicted in the following figure:
10 ATTITUDE REPRESENTATION
2.3
Attitude representation
When we face a problem of attitude representation, the fundamental concept to deal with is the description of the relative orientation between two frames of reference. This task is accomplished by writing each component of the body-fixed reference system in terms of the 3 inertial unit vectors, getting 9 scalar parameters that are usually arranged in matrix form:
A= i1 i2 i3 j1 j2 j3 k1 k2 k3
The matrix A is called attitude matrix or Direction Cosine Matrix (DCM) and represents the rotation from the inertial frame of reference to the body-fixed frame of reference. Defining this matrix is really important since it is characterized by the following useful properties [22]:
– The direction cosine matrix is a proper real orthogonal matrix so its transpose is equal to the inverse matrix:
A−1 =AT (2.1) – The direction cosine matrix representing successive rotations is equal to the multiplication between the attitude matrices relative to the single rotations:
A00 =A0A (2.2) – The matrix A maps any generic vector from the inertial frame of reference onto the body-fixed frame of reference. Given a generic vector vˆI in the inertial
reference system, we have:
AvˆI = i1 i2 i3 j1 j2 j3 k1 k2 k3 v1 v2 v3 = vi vj vk = ˆvB (2.3)
where vˆB is the same vector written in terms of the body-fixed frame of
refer-ence.
The direction cosine matrix doesn’t have a univocal expression, it can be written in terms of different parameters according to what are the needs linked to the problem of attitude representation.
Two classical and very important attitude parameterization are described in the following sections.
2.3.1
Euler Angles representation
The attitude representation in terms of Euler angles is the most classic and is used because it’s easy and immediate to understand its geometrical and physical mean-ing. The concept behind this parameterization is that we can go from a 1st frame
of reference to a2nd frame of reference performing 3 distinct right-hand positive
ro-tations, where the angles characterizing those rotation are exactly the Euler angles. The sequence of rotation is not univocal but can be chosen arbitrary. In this work, a 3-2-1 rotation is considered and it’s performed with the following steps [3]:
1) Rotation around the z-axis of the inertial reference system, represented by cˆ3.
The relative Euler angle φ is defined roll angle.
2) Rotation around the y-axis of the intermediate frame of reference, denoted as
ˆ
c20. The relative Euler angle θ is defined pitch angle.
3) Rotation around the x-axis of the body-fixed reference frame, represented by
ˆi. The relative Euler angle ψ is defined yaw angle.
As said before, each rotation is described by its own direction cosine matrix but, thanks to the property given in (2.2), we can easily calculate the total attitude matrix. The full matrix A is given by:
A321 = cφcθ sφsθ −sθ sθsψcφ−sφcψ cψcφ+sθsψsφ sψcθ sψsφ+sθcψcφ sθcψsφ−sψcφ cψcθ (2.4)
wheres =sin and c=cos.
If we know the matrix A, we can easily calculate the angle θ from:
sinθ=−A1,3 (2.5)
and once we getθ, we can findφandψ exploiting the other relations. This is always possible, except for the case in whichθ is an odd multiple of 90◦ since, in that case, we can only know the sum of the other 2 angles, not their exact values [22].
Due to this singularities problem, different parameterizations of the attitude matrix are sometimes used.
12 ATTITUDE REPRESENTATION
2.3.2
Quaternions representation
The quaternions representation is particularly useful because, describing the attitude through it, we can avoid singularities in our model. However, with this parameteri-zation we lose the immediate physical meaning of the values we’re dealing with and we increase the computational effort. Quaternions are defined as:
q1 =e1sin Φ 2 (2.6) q2 =e2sin Φ 2 (2.7) q3 =e3sin Φ 2 (2.8) q4 =cos Φ 2 (2.9)
Where ˆe and Φ are defined, respectively, Euler axis and Euler angle, and represent the instantaneous rotation axis and the relative rotation angle. From their definition, we can see that the first three components represent a physical vector and that
|q¯|= 1, where q¯= (q1, q2, q3, q4)0.
Applying those parameters for the attitude representation, the direction cosine ma-trix assumes the following form:
A(¯q) = q2 1 −q22−q32+q42 2(q1q2+q3q4) 2(q1q3−q2q4) 2(q1q2−q3q4) −q21+q22−q32+q24 2(q2q3+q1q4) 2(q1q3+q2q4) 2(q2q3−q1q4) −q12−q22+q23+q24 (2.10)
Once we know A, we can get each quaternion component applying the following expressions: q4 =± 1 2[tr(A) + 1] 1 2 (2.11) q1 = A2,3−A3,2 4q4 (2.12) q2 = A3,1−A1,3 4q4 (2.13)
q3 =
A1,2−A2,1
4q4
(2.14)
So, what we basically do is find the4th component of the quaternion and then write
the other 3 in terms of it. Anyway, this is not the only way to compute the 4 components starting from the direction cosine matrix. We have, in fact, 4 possible combinations, where, each different combination, uses a different reference compo-nent of the quaternion. This particular elasticity of the quaternion representation is very useful in order to avoid singularities and increase the numerical computation [22].
As a final summary, the Euler angles representation is very useful for a good under-standing of the physical meaning of the parameters but it can lead to singularities in our computation. The quaternion representation, instead, is good for its elasticity in the calculation of the different components, making us able to avoid singularities. Cons of this parameterization are the higher computational effort and the lack of an immediately understandable physical meaning.
2.4
Attitude kinematics
Since we are dealing with a moving body, it’s clear that the relative position between the frames of reference is not fixed but changes in time and so does, consequently, the attitude matrix. Attitude kinematics is based on the computation of the time derivative of the direction cosine matrix, taking into account the relative angular velocity. Clearly, different attitude representations will also have different attitude kinematics representations.
In case of a representation in terms of Euler angles, attitude kinematics is addressed composing the angular velocities relative to the single rotations. Considering again the rotation 3-2-1 described in section 2.3.1, the total angular velocity is given by [3]:
ω = ˙φc3ˆ + ˙θcˆ02+ ˙ψˆi (2.15) For problems concerning the attitude dynamics and control is clearly interesting and relevant the knowledge of the angular velocity of the body to be controlled, so the angular velocity written in the body-fixed frame of reference. To find it, we have to calculate the components of the vectorial equation(2.15)on the body-axis reference system:
14 ATTITUDE KINEMATICS
ωi = ˙φcˆ3·ˆi+ ˙θcˆ02·ˆi+ ˙ψˆi·ˆi (2.16) ωj = ˙φcˆ3·ˆj + ˙θcˆ02 ·ˆj+ ˙ψˆi·ˆj (2.17) ωk = ˙φcˆ3·ˆk+ ˙θcˆ02·kˆ+ ˙ψˆi·ˆk (2.18)
that leads to:
ωi = ˙ψ−φsinθ˙ (2.19)
ωj = ˙θcosψ+ ˙φcosθsinψ (2.20)
ωk= ˙φcosθcosψ−θsinψ˙ (2.21)
The problems, described before, affecting the Euler angle representation, are also affecting the attitude kinematics.
In terms of quaternions, the description of the attitude kinematics is given, in com-pact form, by:
dA dt = Ω 0 A (2.22) where: Ω0 = 0 ωk −ωj −ωk 0 ωi ωj −ωi 0 (2.23)
We can see that Ω0 coincides with the skew symmetric matrix of the body angular velocity (with sign changed):
Ω0 =−[ω×b ] (2.24) For what concerns the quaternion itself, its time derivative is given by:
dq¯
dt =
1
Where, the matrix Ωhas this shape: Ω = 0 ωk −ωj ωi −ωk 0 ωi ωj ωj −ωi 0 ωk −ωi −ωj −ωk 0 (2.26)
In this work, the equation (2.25) will be applied in a different form, decomposing the vectorial part of the quaternion and the scalar component of the same [36]. Namely:
dq dt = 1 2Ω 0q+1 2q4ωb (2.27) dq4 dt =− 1 2ω T b ~q (2.28) with: q = q1 q2 q3 ; ωb = ωi ωj ωk (2.29)
2.5
Rigid body attitude dynamics
The fundamental equations governing the attitude dynamics of a rigid body are the so called Euler equations, given, in vectorial form, by:
~
M =~L˙ (2.30)
where M~ represents the resulting of external moments acting on the body and
~
L = I~ω represents the angular momentum of the body with respect the center of mass. From now on, talking about angular velocity, it will be always referred to the body angular velocity and it will be indicated simply by ~ω, omitting the pedix b. In the most general case, both the angular velocity and the inertia matrix are time varying so, solving equation (2.30) for the angular acceleration~ω˙, we get the follow-ing expression:
16 RIGID BODY ATTITUDE DYNAMICS
which, in the case of constant inertia matrix simplifies to:
˙
~
ω =I−1(M~ −~ω×I~ω) (2.32) A further simplification of the equation can be performed placing the body-frame in correspondence of the principal axis of inertia, since in that particular frame the inertia matrix assumes this shape:
I = Ix 0 0 0 Iy 0 0 0 Iz (2.33)
So, substituting (2.33) into (2.32), we get:
˙ ωx =Ix−1[Mx+ (Iy −Iz)ωyωz] (2.34) ˙ ωy =Iy−1[My+ (Iz−Ix)ωzωx] (2.35) ˙ ωz =Iz−1[Mz+ (Ix−Iy)ωxωy] (2.36)
where the x-component lies alongˆi, the y-component alongˆj and the z-component along kˆ. In this work, as specified in section (2.2), the body-fixed reference frame is assumed to be coincident with principal axis of inertia so the last simplification described is applied.
In general, the term M~ appearing in Euler’s equations, can be decomposed in 2 different components: external torques (or disturbance torques), Mext, and control
torques,Mc. In the prototype developed, control torques are given by:
– Reaction Control System (RCS)
Control torques are given by 8 cold-gas thrusters and each Euler angle can be modified and controlled.
– Thrust Vector Control (TVC)
Control torques are given by the angular displacement of the engine that cre-ates an off-set between the center of mass and the direction of thrust vector. With this system we can only control pitch and roll angle.
For inputs on the disturbance torques that can affect the vehicle, these may include but not be limited to: torques given by interaction with the atmosphere and by wind gusts, disturbance due to the sloshing of the fuel inside tanks,etc. Sloshing is a phenomenon that occurs in any vehicle experiencing accelerated motion, it is due to the interaction between the fuel and the surface of the tank and can be particularly strong in case of partially filled tanks [28]. This phenomenon is really hard to assess both for its particular dynamics and for the fact that the volume of liquid in tanks is continuously changing in time. In addition, in presence of TVC system, when the mass of the engine is not negligible with respect the total mass, also the relative motion of the engine with respect the body is source of disturbance torques. All of these sources of disturbances should be taken into account in order to avoid instability problems.
2.6
Thrust vector control: literature review
Thrust vectoring is defined as the capability of an aircraft or spacecraft to modify the position of an engine and point the thrust vector in any direction, in order to control the attitude and the angular velocity [31]. This concept has been widely used in many different applications, going from fighter jets to small spacecrafts, and for different purposes. The deflection of the thrust vector is usually obtained by means of different strategies, relative to the type of propulsion system on which the TVC is applied. In general, thrust vectoring methods can be grouped in 3 different categories:
1) Reactive fluid injection 2) Exhaust flow deflection
3) Engine mechanical manipulation
2.6.1
Reactive fluid injection
This method of thrust vectoring is generally applied on rocket engines and consists in injecting a fluid into the side of the flow exiting from a nozzle, by means of particular injectors mounted on the sides of the engine. When the reactive fluid is injected in only one side of the exhaust gas, it changes the properties of the flow, resulting in a different thrust value on that side. This method also increases the axial value of the thrust because of the higher pressure, due to shock waves, and the added mass and energy to the flow. Usually, in order to increase the effect of this system, denser reactive fluid are preferred [14]. An example of the application of
18 THRUST VECTOR CONTROL: LITERATURE REVIEW
this system is the TitanIIIE-Centaur rocket launcher by NASA [1]. Below a scheme of the flow injector on the launcher:
Figure 2.3: Reactive flow injection on TitanIIIE-Centaur (NASA) [1]
2.6.2
Exhaust flow deflection
The method of exhaust flow deflection consists in placing particular surfaces at the exit of the nozzle that can be oriented in order to direct the flow, and consequently the thrust vector, into the desired direction. The most common of those surfaces are external jet vanes and external additional nozzles. This method of thrust vectoring is used both for rocket and jet engines. Jet vanes are external paddles that are placed at the exit of the engine and that can be driven into different configurations in order to guide the exhaust gas. Their main pro is that with them is possible to control each rotation angle, roll included, with just one single engine, not possible for gimbaled engines. A con is that their very presence degrades and reduces the rocket’s efficiency. Vanes are, in general, either designed to be resistant to very high temperature or are constantly cooled down, in order to prevent them from melting. However, lately, some particular composites vanes have been designed to withstand the pressure and temperature of the exhaust flow in the very first part of the flight, until the speed of the body is enough to use normal control surface like aerodynamic fins. Once the control passes to fins, vanes rapidly degrade and dissolve. Those particular vanes are known as dissolvable vanes [11] and are very useful when a lower mass is needed.
A famous example of the application of external vanes are the V-2 rockets, developed during the World War II.
Another solution to obtain the thrust vector tilting by deflecting exhaust gases is the application of an additional external nozzle that is mounted at the exit of the engine. In this way, the desired effect can be obtained by moving the second nozzle while the main engine is kept still.
This category is distinguished from the mechanical manipulation group since, in this particular case, the thrust vectoring is accomplished by the deflection of external components, without moving any part of the engine.
2.6.3
Engine mechanical manipulation
Vehicles exploiting this particular systems are equipped with movable engines or shape-changing nozzle that can be oriented and/or modified to reach the desired goal. By tilting the engine or the nozzle, an off-set between the thrust vector and the center of mass will be generated, giving rise to control torques around the center of mass itself. Those particular systems are usually also referred as gimbaled thrust systems and the angular displacement of the engine is defined gimbal angle. The maximum gimbal angle reachable through the TVC systems defines a cone in which the engine can move, called cone of operation.
Figure 2.4 shows an example of a gimbaled thruster.
Figure 2.4: Example of gimbaled thrust on a rocket [6]
20 THRUST VECTOR CONTROL: LITERATURE REVIEW
rockets but also on aircrafts and spacecrafts. In the first two categories described, thrust vectoring was reached by acting directly on the exhaust flow exiting from the nozzle. In the last case, the flow goes through the nozzle without any modification since the actuation to get the desired deflection is given directly on the engine. When the vehicle is equipped with a single motor, thrust vectoring allows the control only of pitch and yaw angles and has no effect on roll motion (defined as the angular motion around the axial direction of the vehicle). The only way to have a full control on the rotational dynamics of the system through mechanical manipulation is to combine the thrust vectoring in, at least, 2 different engines.
The typical method used to have TVC on jet engines is the usage of adjustable nozzles that can modify their shape in order to assume a deflected configuration, changing so the thrust direction.
In rocket engines, instead, the actuation of the nozzle is usually obtained through the displacement of linear actuators that makes the engine move around a particular joint that fix it to the whole structure. Different types of joint are illustrated below:
Figure 2.5: Typical joints used on gimbaled engines [10]
The gimbal bearing is a joint that allows 2 degrees of motion and that can carry most of the thrust load. The engine is driven by two actuators connected to it, mounted 90◦ apart to provide a full two-axis movement. The gimbal bearing can be replaced by a gimbal ring that allows the same two-axis motion. In case of the usage of a single ring, the actuators will always be mounted 90◦ apart but, in this case, one of them will be attached to the engine and the other one to the ring. We have to underline that 2 actuators are the minimum number to have a full motion
of the engine but usually more than 2 is used for redundancy. The last joint showed in Figure 2.5 is quite different from the others since, in this case, the connection between the engine and the structure is given by a lateral ball and socket joint and the motion is driven by 2 actuators mounted 120◦ apart. This configuration was considered during the Nuclear Engine for Rocket Vehicle Applications (NERVA) program [10]. The main advantage of this configuration is that gimbal bearing and gimbal ring can be avoided. However, the con is that the thrust load must be carried by the lateral connection and by the actuators, making this joint more suitable for low-thrust applications.
Thrust vectoring is commonly used also in spacecrafts and landers. For example, Apollo lunar modules were equipped with a gimbaled engine in order to keep the thrust vector aligned with the center of mass during the gravity turn maneuver in the braking phase of lunar descent [32]. For spacecraft applications, TVC is not always applied on the main propulsion system but also on the thrusters used for attitude control. In [9] is illustrated a spacecraft detumbling procedure using, together with momentum wheels, a reaction control system based on gimbaled thrusters that can adjust their position to get the optimal control torque.
Recalling the LLRV, that vehicle was also equipped with a gimbaled engine but in that case the joint was only used as a gyroscope joint to give an offset between the chassis and the engine, in order to keep the latter always vertical. The aim of this solution was to cancel out 5/6 of Earth gravity acceleration in order to simulate a Moon environment [20].
22 THRUST VECTOR CONTROL: LITERATURE REVIEW
As we see, Thrust Vector Control is applied for different purposes and by means of different configurations and systems. Generation 2 of LEAPFROG is the first one equipped with this technology so an accurate research had been carried out to find the best way to apply TVC on this kind of vehicle. Many of the concepts described before had been analyzed and considered in order to find the best solution that can match with the characteristics of the current prototype and with the ideas planned to be applied in next generations.
Preliminary hardware design
In this chapter are presented and described the considerations and choices made during the design phase of the hardware composing the new prototype, with a particular focus on the hardware dedicated to the TVC system. All the choices have been taken according to the requirements fixed for this project.
3.1
Project budget and requirements
Since a new generation of the project started in September 2019, an entire new prototype had to be designed and developed from scratch, trying to meet as much as possible the initial requirements and goals, stated in chapter 1. The first objectives taken into account were the increase of the payload capacity, the improvement of control and guidance subsystems and the self-reconfiguration capability that the prototype should have. All of this without moving to much from the concept of being a low cost risk reduction platform for simulated lunar flight and landing. In the following sections the different components of the prototype are described according to the decisions made during the design process.
3.2
Structure
The first phase of the design work was focused on the realization of a new structure that could allow an increase off carriable mass, looking forward to a future possible payload application, and an increase in structural resistance. The previous gener-ations, as also shown in figure 1.2, were characterized by a single platform with a 4-leg configuration, a central jet engine and a set of cold-gas thrusters for attitude control. The platform was mainly used just as a storage bed for all the subsystems equipped on the two prototypes while all the load of an eventual free fall would have
24 STRUCTURE
been carried by the legs only. The idea to improve the structure of the new vehicle was to realize a double-layer configuration where two distinct platforms, similar to the ones used in previous prototypes, are connected through an internal chassis. The full CAD design proposed for the new structure is shown below:
Figure 3.1: Structure of Generation 2 - CAD model (front view)
The total height of the structure is 86.6 cm while the total width is about 178 cm. The 4 leg configuration has been applied also in this new prototype.
The two axis in figure 3.2 represent the direction of principal axis of inertia.
The reason for the choice of this particular configuration is that the double layer increases the storage capability, giving the chance to place on board all the sub-systems needed, saving also some space that could be used for an eventual extra payload to test on the vehicle. Moreover, the internal chassis can withstand loads due to a possible free fall, reducing the effort of the legs. Thanks to the usage of carbon fiber, the total mass of the structure is around 8 Kg.
Structural analysis and simulations of the entire model had been performed by other team members in parallel with the design work. The different parts composing the overall structure (chassis, legs and platforms) are described in detail below.
3.2.1
Chassis
The chassis is a reticular structure characterized by two layers connected by transver-sal rods. In order to have a light weight and a high resistance as well, all the tubes used are standard modulus carbon fiber tubes, ideal for bending applications. The rods are connected each other with particular connectors (Dragonplate connectors) designed exactly for this material. Some additional connectors will be 3-D printed in Acrylonitrile Butadiene Styrene (ABS), to match with LEAPFROG’s particular geometry.
(a) Full chassis (top view) (b) Particular of connections between rods
Figure 3.3: Carbon fiber chassis - final version
The red parts visible in figure 3.3 represents the components that are designed to be 3-D printed. The chosen octagonal shape of the chassis is due to manufacturing and high costs issues. The initial designed considered a circular shape to match with
26 STRUCTURE
the geometry of the platforms but that particular solution would have been difficult to realize in labs and too much expensive for a realization in external companies. For this reason, the final choice went to the usage of simple rods connected such to result in a shape as close as possible to the circular one, keeping the overall cost contained.
3.2.2
Legs
The 4 legs are realized with tubes of the same material used for the chassis and each of them is characterized by a main rod connecting the foot to the upper layer of the structure, and two support rods that connect the leg with the lower layer. This particular geometry was chosen to distribute evenly the load among the chas-sis,avoiding a full concentration in a single point. Furthermore, the particular in-clination of the legs was chosen for stability reason but also for possible issues that could be encountered due to the presence of the engine. The engine, in fact, is placed in correspondence of the lower layer of the chassis and the high temperature of the exhaust gases could affect the carbon fiber. An high inclination keeps the legs as far as possible from the engine.
Besides the material used, in order to reduce and dissipate the stress affecting the structure, connection between the leg and the upper part of the chassis comes through a shock absorber while the mounting on the lower layer is made through spring connectors that can displace in presence of a load and then come back to the original position. The 4 feet are realized in memory foam material to further reduce the stress affecting the structure. The connector used to attach the support rods to the main rod will be 3-D printed in ABS material as well.
3.2.3
Platforms
The two platforms connected on the upper and lower layer of the chassis are the storage area for embedded electronics and other subsystems equipped on the proto-type.
Both of them are characterized by a circular shape and a circular hole in the middle. The reasons of the holes are that the lower disc will have the engine going through it while the hole on the upper one ensures that enough air comes to the engine itself. The material used for the realization of the platforms is a Divinycell foam board material stiffened with an epoxy and fiberglass lamination, same procedure used for the realization of the platforms mounted on previous prototypes [37]. The choice for this material is due to its high strength to weight ratio The attachment to the chassis is performed through particular rounded mounting brackets.
3.3
Propulsion system
For the propulsion system, the final choice went to adopt a kerosene powered jet engine by JetCat company, the same type of engine used for generation 0 prototype. The reason is that those engines are easy to manage and doesn’t provide an excessive unwanted yaw rotation due to rotating blade, as a propeller may cause. The problem of the undesired rotation could be eliminated with the installation of a rocket engine, thing planned for future generations.
The generation 0 prototype had installed a JetCat P100 for a much smaller vehicle so, considering that the structure by itself has a mass around 12 kg, the new engine adopted is a JetCat P300 PRO that can provide 200N of thrust more than the previous one. This engine upgrade is necessary since we need a sufficient thrust to carry the structure, all the subsystems installed on it and an eventual payload to test on the prototype, even if the cost of the P300 is quite higher than the P100 one.
Figure 3.4: JetCat P300 PRO [2]
The engine is characterized by a single compressor stage, a combustion chamber and a single turbine stage, and comes with its own telemetry and an embedded Electronic Control Unit (ECU), so they don’t have to be realized from scratch saving time in development. The ECU manages fuel pumps, valves and electronic system thanks to a feedback closed loop that provides back to the engine itself information about RPM of the blades and temperature of exhaust gases. Integrated to the engine there’s also an electric starter used for the initial firing. Other important components of the engine kit are the clamps that are designed and customized exactly for the attachment of the engine to the host structure. The propellant used to feed the engine is store in a dedicated tank placed above the upper platform of the vehicle.
28 REACTION CONTROL SYSTEM
The principal technical specifications of the engine are listed in the following table:
Table 3.1: JetCat P300 PRO technical specifications [2] Basic technical specifications Values
Max RPM 106000
Idle RPM 35000 Thrust at max RPM (N) 300
Thrust at idle RPM (N) 14 Exhaust Gas Temperature (EGT) range (◦C) 480 - 750
Mass flow (kg/s) 0,5 Exhaust gas velocity (km/h) 2160 Fuel consumption at idle (kg/min) 0,143 Fuel consumption at maxRpm (kg/min) 0,784 Weight (kg) 2,73 Diameter (mm) 132 Lenght (mm) including starter 380,5
The term idle used in the table refers to the minimum operating condition.
3.4
Reaction control system
The reaction control system is one of the two attitude control subsystems equipped on LEAPFROG. It is composed by 8 thrusters distributed along the circumference of the upper platform, with the disposition shown in figure:
The two axis depicted in figure 3.5 represent the two in-plane principal axis of inertia (the z axis is perpendicular and coming out the plane). The eight thrusters are placed such that 4 of them (number 2,4,6,8) are used for yaw control and the other 4 (number 1,3,5,7) for roll/pitch control. The reason of the45◦ offset between the thrusters position and the principal axis of inertia is due to the fact that the latter, as shown in picture 3.2, run through the legs of the prototype and placing RCS thrusters above them will result in disturbances due to the interaction between the cold-gas and the legs themselves. Clearly, not having thrusters aligned with principal axis of inertia, we cannot control the rotation angles with a single thruster but we need a combined action of two of them. In particular, the firing logic is the following:
– positive yawing moment → 2 and 6 firing – negative yawing moment→ 4 and 8 firing – positive rolling moment → 1 and 3 firing – negative rolling moment → 5 and 7 firing – positive pitching moment → 1 and 3 firing – negative pitching moment→ 3 and 5 firing
The particular firing logic has been chosen to minimize lateral displacement during attitude control maneuvers. The thrusters are connected, through dedicated pipe lines, to a total of 4 tanks (1 tank for each pair of thrusters) filled with 4000 psi compressed air. The pressure of the air coming out from the tank is tuned by a pressure regulator (one for each pipe line) according to the needs. The air flow coming out from each thruster, instead, is regulated by solenoid valves which open and close according to the signal coming from an in-board CPU receiving data from an Inertial Measurement Unit (IMU) that provides information on the current attitude of the vehicle. The solenoid valves are constituted by an inner coil and a ferromagnetic plug tied, through a spring, into the flow channel, closing it. When the electric signal is sent to the valve, a current starts flowing into the coil and generates a magnetic field that attracts the ferromagnetic plug, opening the channel. Once the electric signal, thanks to the spring, the plug comes back to its standard position, closing the channel again. Those solenoid valves are very useful for this kind of application since their response is pretty fast, opening and closing the channel almost immediately.
30 THRUST VECTOR CONTROL SYSTEM
Figure 3.6: RCS block diagram
3.5
Thrust Vector Control system
The design of the thrust vectoring system has been the main focus of the presented work. This kind of control system is commonly used in lander spacecrafts mainly to keep the thrust vector parallel to the velocity of the spacecraft itself during landing maneuvers, in order to slow down and reach a final soft landing.
For its part, LEAPFROG is meant to be a re-configurable prototype capable to per-form multiple tasks modifying the working configuration of its various subsystems. For the TVC, the guideline has been the design of a system that could be used both as a guidance and navigation system during flight phase and as an attitude control system during hover or landing phases. About those purposes, the design of the attitude control setting is the most delicate since, while guidance and navigation requires slow steering maneuvers and consequently slow changes in thrust vector di-rection, attitude control system has to equilibrate and balance the vehicle every time and it could be called to act against rapid and abrupt attitude variations due to any kind of external disturbance that can affect the dynamics of the system. Although the above described reaction control system, thanks to the fast response given by the solenoid valves, is capable to act rapidly against disturbances, exploiting the thrust
force can reduce the expenditure of energy and cold-gas mass. The main challenge, so, has been the design of a system that could respond to attitude changes as fast as possible, trying to reduce the usage of reaction thrusters. To achieve this goal, since the response of the TVC system, clearly, cannot be infinitely fast, some con-straints and boundary conditions about the range of action had to be fixed, letting the TVC work within some predefined values of thrust deflection angle and adjusting the attitude with the RCS when the eventual required deflection angle exceeds the limits imposed. It is then clear that the synchronization between the two different attitude control system becomes a crucial point for the overall control logic of the prototype. This synchronization, in terms of hardware, is achieved with a merged system of three different CPU’s composed by a main general control unit and two control units dedicated respectively to the RCS and TVC systems, that can work separately and also communicate between each other (a more detailed description about the electronics will be given later on).
The first point to assess was how we could obtain thrust vectoring in our system, considering all the different technologies available in literature. As underlined in section 3.3, the engine chosen for the prototype is an axial jet engine, designed with particular characteristics that cannot be modified or are hard to. For this reason, a solution considering the injection of a secondary fluid was immediately discarded since it would require the modification of the nozzle in order to place injectors around it, proceeding that can be hard and delicate. Moreover, this kind of application requires a precise quantification of the effect of the secondary flow on the main flow, making the deflection of thrust vector hard to assess. The main two options, then, were the deflection of exhaust gases and the mechanical manipulation of the engine. This type of jet engines are characterized by a rigid full body where the nozzle is directly connected with housing for the turbine, the compressor and the combustion chamber. This means that a relative motion of the nozzle with respect the main body of the engine is impossible. This is the main reason for which, on jet engines, is usually applied a deflection of exhaust gases method [25]. Furthermore, those engines are usually applied in light vehicles and the system used to achieve the thrust vectoring should not add an excessive extra mass. As discussed in section 2.6.2 the deflection of the gases coming out from the nozzle is typically achieved using external vanes or an external additional nozzle, systems that are independent from the engine and that allow to keep the latter always fixed to the structure. However, even these applications have some intrinsic complications to deal with during the design process.
Those can be summarized as follow:
32 THRUST VECTOR CONTROL SYSTEM
in contact with the exhaust gases that, as shown in table 3.1, can reach a tem-perature around 700 ◦C so the material should be high-temperature resistant, aspect that could lead to an expensive choice.
2) Engine performance variation: When a system is acting at the exit of an engine nozzle, it unavoidably affects and changes the specifications and performance of the engine itself. For example, the presence of external components reduces the velocity of the exhaust flow and consequently the thrust provided by the engine, being the two quantity directly proportional, as underlined by the thrust equation [26]:
F = ˙meue+ (Pa−Pe)Ae (3.1)
where F represents the thrust vector and ue the velocity of the flow in the
di-rection of vehicle’s motion. The adoption of those systems necessarily requires a deep testing phase where the variations of the engine specifications must be quantified and addressed.
For this reasons, combined to the fact that we can allow a little increase of the vehicle mass, the final decision went to reach thrust vectoring by mechanical manipulation of the engine. Moreover, applying this concept, the same control logic used to drive the thrust vector in this prototype can be re-used in next generations also in case of the installation of a mono-propellant rocket engine as propulsion system.
Since, as stated before, in JetCat P300 the nozzle is fixed to the main body, the thrust vectoring can be reached only through mechanical manipulation of the entire engine. Concerning this, a simple gimbal joint cannot be used with this type of engine since the upper part must be free to allow the air to enter the compressor, so, the final idea to get the adjustability of thrust vector direction was to gimbal the engine through a gimbal ring.
A gimbal ring is a gyroscopic joint composed by a set of concentric rings that can rotate with respect each other, along different directions. This particular joint, when connected around a body, allows the body itself to tilt in any wanted direction. In our particular application, the final design of the gimbal ring provides three concentric rings where the inner one is directly connected to the engine and the outer one is attached to the chassis. The connection between the inner ring and the engine is obtained through particular clamps that are custom designed exactly for this engine and provided by the same company. Thanks to the three ring configuration we get one degree of freedom with the relative rotation of the inner ring with respect the middle ring and another degree of freedom with the relative rotation of the
middle ring with respect the outer one. The rotations are allowed by the presence of particular pins that interconnect the ring and that, at the same time, have the role of axis of rotation.
Figure 3.7: Conceptual sketch of the gimbal ring
As we can see from figure 3.7, the gimbal’s axis of rotation are perfectly aligned with the principal axis of inertia of the vehicle (x and y in the figure). The choice is related to a simpler actuation procedure since, in case of pure pitch or pure roll motion, we can provide the control torque by rotating only one of the rings so, as will be described later on, activating only one linear actuator.
Once decided the method to use to obtain the TVC, the second phase of the work regarded the mechanical design of the gimbal ring.
3.5.1
Gimbal joint design
Gimbal rings are very particular joints that are custom designed for the applications they’re intended for. The particular geometry of the prototype, combined with the engine used, required a newly designed joint. The design of the gimbal ring had been driven by some requirements that can be summarized as follows:
• Overall component as light as possible, so not to add a too much extra mass to the vehicle
• Enough resistance in order to withstand the thrust given by the engine
• Use of COTS (Commercial Off The Shell) material, in order to keep the cost sufficiently low
34 THRUST VECTOR CONTROL SYSTEM
• Not too complicated geometry in order to allow an in-lab machining and fab-rication with the machinery available.
In order to meet the different requirements, the final choice for the material went to usage of 3 mm thick aluminum 6061-T6 sheets. This particular material is widely used, also in aerospace applications, for its good machinability, medium to high strength and light weight, all of that at an affordable price [24]. The only components of the gimbal ring with a different material are the inter-connecting pins since they’re carrying both the weight of the rings and both the thrust of the engine. For this reason, the pins are characterized by a stainless steel screwed shaft where a couple of bushings are placed in order to facilitate the rotation. The distance between rings is kept through the usage of spacers. Bushings and spacers are made in stainless steel too. Concerning the geometry, the first and most intuitive idea was the realization of simple rounded shape rings to connect with pins. However, this geometry had some issues in terms of attachment to the structure and, primarily, in terms of manufacturing. A round shape, in fact, requires a very precise bending of the aluminum sheet, hard to achieve with the available machines. Moreover, just because the gimbal ring is made out from aluminum sheets, we could not get a full body ring. For this reason, the final decision went to build the gimbal joint by assembling together different aluminum plates, easy to realize with the available equipment. The final shape recalls the octagonal geometry of the lower layer of the chassis in order to get an easier attachments between the two parts. In addiction, the inner ring of the joint also hosts two vertical plates that run along the sides of the engine and that are used as a connection point for the actuators.
Final design of the gimbal ring mounted on the engine:
(a) Rest condition (b) Displaced condition
3.5.2
Actuation system
As briefly mentioned before, while the gimbal ring allows the motion of the engine, the actual movement is obtained thanks to the action of two linear actuators that are placed 90◦ apart and that act at the same time on the engine. In particular, they are placed such to be aligned with principal axis of inertia of the prototype. In this way, a pure pitch or roll rotation is adjusted with the displacement of a single actuator.
The actuators employed, in order to save money, were not chosen from the market but, instead, they were some remaining actuators available at SERC from a different project. This aspect was challenging since, being those not properly designed for this system, they’re work had to be optimized in order to get the best performance out of them.
Going into detail, they are industrial linear electro-mechanical actuators from Win-dynation company (WinWin-dynation LIN-ACT1-04). The sketch is given below:
Figure 3.9: Windynation linear actuator [33]
The main technical specification are listed in the following table:
Table 3.2: Windynation LIN-ACT1-04 technical specifications [33] Basic technical specifications Values
Stroke Length 102 mm Rated Load 900 N Travel Speed (Max load) 10 mm/s
Rated Voltage 12 VDC Current Draw (Max load) ≤25A
Operating Temperature -26◦C to 65◦C
36 THRUST VECTOR CONTROL SYSTEM
Looking at table 3.2, between the different specifications, the most important for our application are the rated load, the travel speed, the stroke length and the duty cycle. All those parameters had to be taken into account because they strongly affect the performance of our control system. For example, the duty cycle parameter shows that actuators can only work for 25% of the time, with a 75% of non working condition. This shows that a full TVC-based control system is hard to have, even if attitude control, in particular situation, can also request just few seconds of action. The problem of the duty cycle could be avoided with the usage of hydraulic actuators. However, electro-mechanical actuators are usually preferred because they represent a lighter, safer and easier to maintain solution for this kind of work [8]. While duty cycle affects the working time of the actuators, the other parameters directly affect the performance of the latter, in particular the speed of actuation. Luckily, those actuators have not to carry a very high load since the engine has a weight of 2,73 kilograms and it is totally sustained by the gimbal ring. Actuators, so, only have to face the inertia of the engine itself. This means that their velocity is much higher than the one given in table 3.2. The manufacturing company provides the following graph showing the relationship between the carried load and the speed of actuation.
Figure 3.10: Actuators: speed vs load [33]
As we can see, in case of load-free movement, the speed of the actuators increase up to 35,6 mm/s (1,4 inches/s). Taking into account the same considerations made before about the weight of the engine, we can assume that the actuators work at a velocity close to the maximum one. Introducing a sort of "safety factor", in
the present work is assumed that the actuators work at a speed of 30 mm/s (1.18 inches/s). However, in future work, a test of the actuators on the engine fired should be performed in order to assess the real working speed. Actuator’s performance can be further improved optimizing the placement of the actuator itself with respect the engine. Typical applications of the TVC on aircraft or rockets, because of the particular geometry of the engines, are characterized by an inclined position of the actuators driving the engine. However, this particular working condition is limiting the actuators performance for two main reason:
1) When the actuator is inclined with respect the gimbal joint, its displacement does not contribute entirely to the movement of the engine. In fact, only the component perpendicular to the joint is actually acting on the engine displacement while the parallel component is not useful for the purpose. The actuation speed is then reduced since a larger displacement is required to achieve a desired point.
2) Inclined actuators means that their pivot point is not perpendicular to the flying direction. This condition results in a strong coupling in the load motion between pitching and yawing directions, making more difficult the coordination between the two actuators [18].
In our application, actuators should be as fast as possible in order to have a good controllability of the attitude through the TVC. Taking into account what explained before, thanks to the particular position in which the engine is placed, the actuators, on LEAPFROG, are placed perfectly perpendicular to the axis of motion, such to have the full displacement of the same contributing to drive the engine in the desired position.
A further improvement is achieved minimizing as much as possible the distance between the gimbal point, i.e. the point around which the rotation of the engine occurs, and the point in which the single actuator acts on the engine itself. As already pointed out, the key behind the Thrust Vector Control is the creation of an offset between the center of mass of the vehicle and the direction of the thrust vector by inclining the engine. When we talk, then, about a fast actuation, what is meant is that we should be able to incline the engine as fast as possible. Reducing the distance between the center of rotation and the actuator acting point helps in this purpose since, for the same amount of linear displacement, with a shorter distance a bigger angular displacement of the engine is induced and so we increase the tilting speed.
The installation configuration of the actuators on LEAPFROG is depicted in figure 3.13.
38 THRUST VECTOR CONTROL SYSTEM
Figure 3.11: Actuators installation scheme
As shown in the picture, in our practical case the distance between the two point is about 9 cm. From this data, combined with the assumed actuators speed of 3 cm/s, we can see that the angular actuation speed of this system is around 0.3218 rad/s (about 18.4◦/s). However, even if the actuation speed is quite satisfying, the angular motion of the engine is limited by different factors like the interference with the chassis and the restricted movement of the actuators due to the presence of the platform. For this reason, a limited work range for the gimbal system has been fixed. In particular, if we defineGimbal offset angle δthe angle between the vertical direction and the current thrust direction, the range of values that this angle can assume has been limited to:
0◦ <|δ|<5◦ (3.2) This means that the cone of operation has an opening of 5◦. In this way, the inter-ference problems are avoided and, considering the angular actuation speed specified above, we can have a fast actuation in the whole work range since the entire angular displacement (10◦) is covered in less than one second.
The main issue that can arise with this placement of the actuators is their relative vicinity with the exhaust gases of the engine. The maximum working temperature allowed for the actuators, as specified in table 3.2, is 65◦C while the exhaust flow coming out of the nozzle can reach temperature around 750◦C. For this reason a thermal protection for the actuators may be needed.
3.5.3
Embedded electronics
The full embedded electronics system designed to drive and command all the differ-ent compondiffer-ents of the TVC is given in the following block diagram:
Figure 3.12: Thrust vectoring system embedded electronics
The red and black cables in the diagram represent an electric connection in which, in particular, red cables represent the positive pole and black cables represent the negative one. Blue cables, instead, represent a data connection between the different blocks.
The central core of the electronics system is represented by the control board, an Odroid CPU that exchanges information with LEAPFROG’S main CPU and IMU sensor. When the main CPU commands a variation of thrust vector amount, the control board feeds the information to the engine ECU that manages the engine performance. The power needed by the JetCat ECU and by the control board is furnished by a couple of LiPo batteries. In the case of the control board, the power coming from the battery is first processed by the UBEC block, a voltage regulator that feeds to the CPU a constant power with the proper values of voltage and current intensity. In the case, instead, in which the main LEAPFROG’S CPU commands an attitude adjustment maneuver, the control board, this time, communicates with the transistors that manage the on-off switching of the actuators. The transistors are identified by the 2 blocks LM338T Voltage control. Those components, in fact, according to the signal received by the control board, change the sign of the voltage given to the actuators, triggering the forward motion or the backward motion of the latter. The power needed by the actuators is furnished by a third LiPo battery. A current sensor is placed in the circuit in order to know the amount of current that is actually flowing. The battery used to power the actuators affects their performance
